KBCC The History of Math and Ancient Greek Mathematics Worksheet

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CUNY Kingsborough Community College

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Homework Chapter 2: Babylonian Mathematics Babylonian Arithmetic Problems 1) Convert the following numbers into base 60. 78, 126, 38921, 1432, 216,061, 42532 2) a) b) c) d) e) f) Convert the following base 60 numbers to decimal numbers 1, 12 3, 15, 11 1, 1, 1 4, 0, 3 1, 4 1, 0, 4 3) Translate the numbers in the following table to decimal numbers. Here are some Babylonian Algebra and Geometry problems. Note that although we refer to them as “algebra” problems, they didn’t have algebra as we know it. Nevertheless, they are quite impressive. Solve only the circled problems. Chapter 3 Homework: Ancient Greek Mathematics - Part I 1. Use the figure below to calculate the height of the Great Pyramid like Thales did. 2. The sum of two consecutive triangular numbers is a square number. Give a “proof -bypicture.” 3. The sum of the first n odd numbers is n 2 . Give a “proof -by-picture” and explain it. 4. The sum of the first n numbers is n ( n + 1) . Give a “proof -by-picture” and explain it. 2 5. Plutarch (c 100 AD) stated that if a triangular number is multiplied by 8, and 1 is added, then the result is a square number. Give a “proof-by-picture” and explain it. 6. In 1636 Fermat conjectured in a letter to Mersenne that “every positive integer was the sum of three or fewer triangular numbers. In 1665, Pascal wrote a book called Treatise on Figurative Numbers where he presented this conjecture. It was first proved by Gauss in 1801. Write each of the following numbers as the sum of three or fewer triangular numbers 16, 25, 39, 56, 69, 150, 185, 287. 7. 8. Give a “proof by picture” of the Pythagorean Theorem from the Zhoubi Suanjing and explain it. The MAA has a Catalog of Mathematical Treasures and the above image is taken fromit. This is another good link. 9. Give a “proof by picture” of the Pythagorean Theorem from the Vigaganita and explain it. 10. Use similar triangles to prove the Pythagorean Theorem (from the Elements Book VI,). 11. 12. The Pythagorean Problem states “find all natural number solutions of the Pythagorean equation x 2 + y 2 = z 2 .” A triple ( x, y, z ) of natural numbers satisfying the Pythagorean equation is called a Pythagorean triple. a) Verify that (3, 4, 5) is the only Pythagorean triple involving consecutive positive integers. b) Verify that x = 2n + 1 , y = 2n 2 + 2n and z = 2n 2 + 2n + 1 for n  1 satisfies the equation. Pythagoras gave this solution. Discuss how he may have obtained it. c) Verify that x = 2n , y = n 2 − 1 and z = n 2 + 1 for n  1 satisfies the equation. Plato gave this solution. Discuss how he may have obtained it. d) Verify that x = 2mn , y = m 2 − n 2 and z = m 2 + n 2 for m, n  1 satisfies the equation. Euclid states this in Book X and conjectures that it is a complete solution. It was eventually proved by the Arabs. We saw this in the section on Plimpton 322. Give the proof we discussed in that section. 13. State the three construction problems from antiquity. What was Platos restriction on the problems? How were they resolved? 14. Explain how to double the square. 15. Explain how to bisect an angle. 16. Explain how to trisect a a right angle. 17. Describe Ahmes’ method for squaring the circle. Homework Chapter 4: Ancient Greek Mathematics – Part II 4.1. Euclid’s Elements 1. Proposition I.1. is a construction problem. It states “Given a segment AB, draw an equilateral triangle on it.” Give Euclid’s proof and the flaw in it. How can the proof be fixed? 2. Proposition I.4. is the famous SAS property. It states “Two triangles are congruent if two sides and the included angle of one are congruent to the corresponding sides and the included angle of the other.” Describe the flaw in this proof and explain what can be done about it. 3. Proposition I.5. states “In triangle ABC, if side AB equals side AC, then angle ABC equals angle ACB.” Proposition 6 is the converse of Proposition 5. It states “In triangle ABC, if angle ABC is equal to angle ACB, then side AB equals side BC.” Prove Propositions 5 and 6. 4. Proposition I.16. is the exterior angle theorem. It states “The exterior angle of a triangle is greater than either opposite interior angles.” Give Euclid’s proof and the flaw in it. How can the proof be fixed? 5. Prove Proposition I.27. states “If a transversal cuts two lines making equal alternate interior angles, then the lines are parallel.” 6. Proposition I.32. in Book I states “The sum of the angles of a triangle equal 180.” Prove it 7. Proposition I.47. is the Pythagorean Theorem. In a right-angled triangle the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Give Euclid’s “Mousetrap” proof. 8. In Book X, Euclid defined two segments to be commensurable if they can be measured by the same measure and incommensurable otherwise. Explain what he meant by this. 9. Proposition IX 20. states that there are infinitely many primes. Prove it. 10. Proposition X 9. states that 2 is irrational. Prove it. 4.2. Euclid’s Number Theory 1) State Euclid’s Division Algorithm 2) State Euclid’s algrithm for finding the greatest common divisor of two numbers. 3) Prove the following statements: a) Two natural numbers a and b are relatively prime if and only if there exist integers x and y such that ax + by = 1 . b) If two natural numbers a and b are relatively prime and a | c and b | c , then ab | c . c) If p is prime and p | ab , then p | a and p | b . 4) State the Fundamental Theorem of Arithmetic 6) Use the Sieve of Eratosthenes to find all primes up to 200. 4.3. Archimedes 1) The next set of questions is based on the Chapter 4 (Archimedes) from Dunham. This is an excellent example of the kind of paper you should be writing. It has interesting stories and a great theorem. a) Consider a regular polygon with area A, perimeter Q and apothem h. (Apothem is the length of the line drawn from the polygon’s center and perpendicular to one of the sides. Prove that area of a regular polygon is ℎ𝑄 𝐴= 2 b) Prove Euclid’ Proposition IV(6). Given a circle we can inscribe within it a square. This is one of Euclid’s constructions. Euclid presents it as “To inscribe a square inside a circle.” c) Great Theorem: Prove that the area of a circle is equal to a right-angled triangle, in which one of the sides about the right angle is equal to the radii, and the other to the circumference of a circle. This is the first of three results in Archimedes’ paper “Measurement of a circle.” d) Explain how (c) leads to the area of a circle formula 𝐴 = 𝜋𝑟 2 . Chapter 5: Ancient Greek Mathematics – Part III 1) The first set of questions is based on pages 11 – 20 from Dunham on Hippocrates’ Quadrature of the Lune. a) b) c) d) e) f) g) What does quadrature of a plane figure mean? Prove that the rectangle is quadrable. Prove that the triangle is quadrable Prove that a 5-sided polygon is quadrable Explain how (d) can be extended to an n sided polygon. What is a lune? Prove that a lune is quadrable The purpose of this sort of assignment where you have to read and learn the math on your own is to prepare you for researching and writing the term paper. If you were to choose Hippocrates as your subject, then you would begin by giving a description of his life, some interesting events perhaps, and then launch into an analysis of the mathematics, sort of like Dunham has done in this extract. His main theorem is the quadrature of the lune and it is considered a “great theorem.” 2) Diophantus said “when you are acquainted with what I have presented, you will be able to find the answer to a many problems which I have not presented, since I shall have shown to you the procedure for solving a great many problems and shall have explained to you an example of each of their types.” a) Continuing the Babylonian tradition, Diophantus asks in Book I #27 for two numbers whose sum is 30 and product is 96. Find the numbers. b) Find two square numbers having a difference of 60 (Book II #10) c) Divide a square, say 64, into the sum of two squares. (Like Book II #8) d) Find a number, such that if two numbers say 8 and 9 are subtracted from it, both remainders are squares. (Like Book II #13) e) Find a number that can be added to two numbers, say 5 and 6, so as to make each of them a square. (Like Book II #11)
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2} to form a triangular number we begin with 1

We say 1 is the first triangular number, to form the next we add gnomon 2:

So 3 is the second consecutive triangular number
The sum of the two consecutive triangular numbers
=1+3=4
To proof 4 as a square number by picture

3} if the nth odd number is 2n-1
So we prove that
1+3+…+ (2n-3) + (2n-1) =𝑛2
It can be proved using arithmetic progression
Sum of an A.P with n terms, difference and last term
𝑛

With the formula 𝑠𝑛 = 2 (𝑎 + 𝑙)
=𝑡1=𝑎 , 𝑡2=𝑎+2𝑑… 𝑡𝑛=𝑎+(𝑛−1)𝑑
𝑛
2

=𝑠𝑛 = (𝑎 + 𝑎 + (𝑛 − 1)𝑑
𝑛

=𝑠𝑛= 2 (𝑎 + 𝑡𝑛 )
Applying this to the series
𝑛

S=2 (1 + 2𝑛 − 1)
𝑛

=2 (2𝑛) = 𝑛2
4}
5} using the first triangular number 1

Multiplied by 8 =1*8=8

Add 1
=
Nine is a square number: proof

6}Triangular numbers =1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190…etc.
16=10+6
25=10+15
39=21+15+3
56=55+1
69=45+21+3
150=105+45
185=120+55+10
287=190+91+6
1
𝑎

1


1


7a} − = −

1
𝑏

Finding the LCM at both sides
ℎ−𝑎

= 𝑎ℎ =

𝑏−ℎ
𝑏ℎ

Cross multiply
=𝑏ℎ ∗ (ℎ − 𝑎) = 𝑎ℎ ∗ (𝑏 − ℎ)
ℎ−𝑎

𝑎ℎ

ℎ−𝑎

𝑎

=𝑏−ℎ = 𝑏ℎ
=𝑏−ℎ = 𝑏

Therefore h satisfies the value of harmonic means of a and b in this relation
2𝑎𝑏

7b} ℎ = 𝑎+𝑏

Cross multiply
=h (a+b) =2ab
= ha+hb=2ab
This proves that h does not satisfy its value for harmonic mean of a and b
8} the area of the four bold triangles is equal to the area of the large, oblique square minus the small
square in the center.
9}
1

10} Area of the trapezoid (EBCD) = 𝐴 = 2 (𝑎 + 𝑏)(𝑎 + 𝑏)
=

(𝑎+𝑏)2
2
1

1

1

Area of the the three triangles (ABC+EAD+EBA) = 𝐴 = 2 (𝑎 ∗ 𝑏) + 2 (𝑐 ∗ 𝑐) + 2 (𝑎 ∗ 𝑏)
=

𝑎𝑏
2

+

𝑐2
2

+

𝑎𝑏
2

= 𝑎𝑏 +

𝑐2
2

Since area of trapezoid=area of 3 triangles
=

(𝑎+𝑏)2
2

𝑐2

= 𝑎𝑏+ 2

=(𝑎 + 𝑏)2 = 2𝑎𝑏 + 𝑐 2
=𝑎2 + 𝑏 2 + 2𝑎𝑏 = 2𝑎𝑏 + 𝑐 2
=𝑎2 + 𝑏 2 = 𝑐 2 𝑝𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑡ℎ𝑒𝑜𝑟𝑒𝑚
12a} (3, 4, 5)
=any other Pythagorean triple are common multiples of the above Pythagorean theorem
To get another valid Pythagorean triple (a,b,c)
(k*a ,k*b ,k*c) where k is a positive integer
E.g. Let k =2
Then (2*3,2*4,2*5)
=(6,8,10) which is a Pythagorean triple
:. The triples cannot be consecutive positive integers
Hence (3,4,5) is the only Pythagorean triple

12b} 𝑥 = 2𝑛 + 1, 𝑦 = 2𝑛2 + 2𝑛, 𝑧 = 2𝑛2 + 2𝑛 + 1 𝑓𝑜𝑟 𝑛 ≥ 1 …equation 1
Pythagoras arrived at the above solution from the relation:
(2𝑘 − 1) + (𝑘 − 1)2 = 𝑘 2 …equation 2
And then searching for those k for which 2k-1 is a perfect square, I.e., 2k1=𝑚2 (𝑠𝑖𝑛𝑐𝑒 𝑚2 𝑖𝑠 𝑜𝑑𝑑, 𝑚 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑜𝑑𝑑). 𝑡ℎ𝑖𝑠 𝑔𝑖𝑣𝑒𝑠:
=𝑘 =

𝑚2 +1
2

𝑎𝑛𝑑 𝑘 − 1 =

𝑚2 −1
2

Thus the relation (2) can be written as:
= 𝑚2 +

(𝑚2 −1)2
2

=

(𝑚2 +1)

2

2

From which it is clear that (1) is satisfied with:
𝑚2 �...


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Great study resource, helped me a lot.

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